Shintani relation for base change: unitary and elliptic representations

• Main Authors: Badulescu, A. I, Henniart, G
• Format: Journal Article
• Language: English
• Published: 23.05.2016
• Subjects: Mathematics - Number Theory
• DOI: 10.48550/arxiv.1605.06948

Let $E/F$ be a cyclic extension of $p$-adic fields and $n$ a positive integer. Arthur and Clozel constructed a base change process $\pi\mapsto \pi_E$ which associates to a smooth irreducible representation of $GL_n(F)$ a smooth irreducible representation of $GL_n(E)$, invariant under $Gal(E/F)$. When $\pi$ is tempered, $\pi_E$ is tempered and is characterized by an identity (the Shintani character relation) relating the character of $\pi$ to the character of $\pi_E$ twisted by the action of $Gal(E/F)$. In this paper we show that the Shintani relation also holds when $\pi$ is unitary or elliptic. We prove similar results for the extension $C/R$. As a corollary we show that for a cyclic extension $E/F$ of number fields the base change for automorphic residual representations of the ad\`ele group $GL_n(A_F)$ respects the Shintani relation at each place of $F$.